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11 votes
2 answers
1k views

Class groups of normal domains over finite fields

Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be ...
Hailong Dao's user avatar
  • 30.6k
10 votes
2 answers
1k views

Does a universal Frobenius map exist?

For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p. Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ...
Marc Nieper-Wißkirchen's user avatar
12 votes
2 answers
1k views

Weil Conjectures for nonprojective algebraic varieties

If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?
John McCarthy's user avatar
5 votes
2 answers
1k views

Weil Conjectures for Grassmannians

To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
John McCarthy's user avatar
10 votes
2 answers
393 views

Counting points on varieties of low codimension

The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...
David E Speyer's user avatar
8 votes
1 answer
637 views

Cohomology map induced by the group actions on homogeneous vector bundles

Here is a topological question which seems quite elementary. The answer to this question may be useful e.g. in estimating the orders of the automorphism groups of some algebraic varieties and in ...
algori's user avatar
  • 23.5k
15 votes
2 answers
2k views

How Does a Borel Subgroup Know Which Weights Are Dominant

Let $G$ be a simple group (say $SL_n$) and let $B$ be a Borel subgroup (say upper triangular matrices). Then all irreducible representations of $G$ are induced from one-dimensional representations of $...
Dinakar Muthiah's user avatar
12 votes
1 answer
513 views

Littlewood–Richardson–Type Rule for Cohomology Ring of Grassmannians

$\DeclareMathOperator\GL{GL}$The ordinary Grassmannian of k-planes in n-space is a coset space for $\GL_n$. It is $\GL_n$ mod a maximal parabolic. Here there is a nice basis given by Schubert ...
Dinakar Muthiah's user avatar
15 votes
5 answers
3k views

Can we count isogeny classes of abelian varieties?

Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...
Rebecca Bellovin's user avatar
17 votes
2 answers
1k views

Can Hom_gp(G,H) fail to be representable for affine algebraic groups?

Let $G$ and $H$ be affine algebraic groups over a scheme $S$ of characteristic 0 and let $\textbf{Hom}_{S,gp}(G,H)$ be the functor $T \mapsto \text{Hom}\_{T,gp}(G,H)$ Theorem (SGA 3, expose XXIV, 7....
David Zureick-Brown's user avatar
13 votes
3 answers
816 views

Constructing a degeneration (as a group scheme) of G_m to G_a

SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga. What is such an example?
David Zureick-Brown's user avatar
2 votes
1 answer
276 views

Do subgroups respect the orbit-closure relation?

Suppose G is a Lie group (or algebraic group) acting on a manifold (or scheme) X, and H⊆G is a subgroup. Let x,y∈X be points such that x is in the closure of the orbit H⋅y (but not in H&...
Anton Geraschenko's user avatar
1 vote
2 answers
325 views

How to make commutative algebraic groups strongly dualizable?

Let's use the notation of [A=>B] for Hom(A, B). Take a 1-dimensional algebraic torus G<...
Ilya Nikokoshev's user avatar
11 votes
4 answers
3k views

What does ramification have to do with separability?

Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
David Corwin's user avatar
  • 15.4k
48 votes
5 answers
15k views

Algebraically closed fields of positive characteristic

I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
Harrison Brown's user avatar
42 votes
4 answers
8k views

Tannakian Formalism

The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ...
Dinakar Muthiah's user avatar
13 votes
2 answers
3k views

Is the fixed locus of a group action always a scheme?

Suppose $G$ is an algebraic group with an action $G\times X\to X$ on a scheme. Does the fixed locus (the set of points x∈X fixed by all of $G$) have a scheme structure? You can obviously define the ...
Anton Geraschenko's user avatar
14 votes
2 answers
989 views

Do orbits and stable loci of group actions have natural scheme structures?

Suppose G is an algebraic group with an action G×X→X on a scheme. Then many of the usual constructions you make when you talk about group actions on sets can be made scheme-theoretically. ...
Anton Geraschenko's user avatar
11 votes
4 answers
2k views

Constructing Affine Kac-Moody Groups

Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the ...
Dinakar Muthiah's user avatar
9 votes
3 answers
2k views

Characterisation for separable extension of a field

Can someone verify this for me.. or tell me what reference shows me this... is this true: Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
Jose Capco's user avatar
  • 2,275
8 votes
2 answers
481 views

Division Algebras as Algebraic Groups

If I'm given a division algebra D with Z(D)=F, then how can I view Dx as an algebraic group defined over F? I'd like to see first how Dx can be given the structure of a variety defined over F, and ...
Joel Dodge's user avatar
  • 2,799
16 votes
3 answers
5k views

What is an Oper?

Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of G-local systems. I keep reading that there's a substack of "opers" but am having trouble locating a definition....
Charles Siegel's user avatar
7 votes
3 answers
2k views

Iwasawa and Cartan Decompositions.

Consider the tome of Bruhat and Tits: Groupes réductifs sur un corps local : I. Données radicielles valuées. Publications Mathématiques de l'IHÉS, 41 (1972), p. 5-251. (available on NUMDAM). I am ...
Peter McNamara's user avatar
15 votes
3 answers
4k views

Iwasawa Decomposition

Does anyone know where I can find a proof of the Iwasawa decomposition for reductive groups? I know that there are a couple of related results that are called the Iwasawa decomposition, but I am ...
Dinakar Muthiah's user avatar
18 votes
7 answers
6k views

Langlands Dual Groups

Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does ...
Charles Siegel's user avatar
19 votes
2 answers
1k views

Hopf algebra reference

I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf ...
David E Speyer's user avatar
14 votes
6 answers
2k views

Does every morphism BG-->BH come from a homomorphism G-->H?

Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, ...
Anton Geraschenko's user avatar
31 votes
7 answers
10k views

Quotients of Schemes by Free Group Actions

I've often seen people in seminars justify the existence of a quotient of a scheme by an algebraic group by remarking that the group action is free. However, I'm pretty sure they are also invoking ...
Dinakar Muthiah's user avatar
9 votes
3 answers
1k views

Unipotent linear algebraic groups

Let $U_1$ be a unipotent group inside some Chevalley group $G$. For now, think of $G$ as being $SL_n(K)$ where $K$ is a field; then we can take $U_1$ to be a bunch of strictly upper triangular matrics....
Nick Gill's user avatar
  • 11.2k
22 votes
4 answers
4k views

Is there a "universal group object"? (answered: yes!)

I want to say that a group object in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One ...
Andrew Critch's user avatar
8 votes
2 answers
8k views

What does "supersingular" mean?

Are supersingular primes and supersingular elliptic curves related? (this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
Ilya Nikokoshev's user avatar
7 votes
1 answer
718 views

Ways to characterize supersingular primes?

I've read the definition, and it basically says p is a supersingular prime iff the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational. And there's a ...
Ilya Nikokoshev's user avatar
79 votes
12 answers
13k views

Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
Qiaochu Yuan's user avatar
27 votes
7 answers
6k views

Etale covers of the affine line

In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
Tyler Lawson's user avatar
  • 52.7k
6 votes
4 answers
1k views

When is a map given by a word surjective?

Let $w(x,y)$ be a group word in $x$ and $y$. Let $x$ and $y$ now vary in $\operatorname{SL}_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of ...
H A Helfgott's user avatar
  • 20.2k
25 votes
4 answers
2k views

algebraic group G vs. algebraic stack BG

I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
Anton Geraschenko's user avatar
10 votes
5 answers
990 views

Non-conjugate words with the same trace

Let n>=2, p a large prime, G = SL_n(Z/pZ). If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
H A Helfgott's user avatar
  • 20.2k
30 votes
5 answers
4k views

Deformation theory of representations of an algebraic group

For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that the obstruction to deforming V as a representation of G is an element of H2(G,V&...
Anton Geraschenko's user avatar
20 votes
5 answers
4k views

Equivalent statements of the Riemann hypothesis in the Weil conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
Brandon Levin's user avatar
13 votes
3 answers
1k views

How to topologize X(R) when R is a topological ring?

Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...
Brandon Levin's user avatar
30 votes
2 answers
10k views

When is fiber dimension upper semi-continuous?

Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$. When is this function ...
Anton Geraschenko's user avatar
42 votes
9 answers
6k views

Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?

Harold Williams, Pablo Solis, and I were chatting and the following question came up. In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
Anton Geraschenko's user avatar
37 votes
4 answers
12k views

Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element? Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
Anton Geraschenko's user avatar

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